in Xin [1]}. In fact, two estimations (REF ) and
\((\ref {w1})^{\prime }\)  are equivalent for fixed wave strength \(w_+-w_-\) .
However, the advantage of Lemma REF  is that the energy
estimate can be carried out for all time since there is no
singularity to the approximate rarefaction wave even at \(t=0\) . While
in Xin's paper [1]}, the energy estimate must be done in two
time-scalings, that is, finite time and large time, due to the
singularity of the estimations of the approximate rarefaction wave
at \(t=0\) .
